Analysis by its history:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer
2008
|
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. [358]-368) and index |
| Beschreibung: | x, 377 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9780387770314 9780387770369 |
Internformat
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Datensatz im Suchindex
| _version_ | 1805092133685690368 |
|---|---|
| adam_text |
Contents
Chapter I Introduction to Analysis of the Infinite
1.1 Cartesian Coordinates and Polynomial Functions
. 2
Algebra
. 2
"Algebra Nova"
. 6
Descartes's Geometry
. 8
Polynomial Functions
. 10
Exercises
. 14
1.2
Exponentials and the Binomial Theorem
. 17
Binomial Theorem
. 18
Exponential
Función
. 25
Exercises
. 28
1.3
Logarithms and Areas
. 29
Computation of Logarithms
. 30
Computation of Areas
. 33
Area of the Hyperbola and Natural Logarithms
. 34
Exercises
. 39
1.4
Trigonometric Functions
. 40
Basic Relations and Consequences
. 43
Series Expansions
. 46
Inverse Trigonometric Functions
. 49
Computation of Pi
. 52
Exercises
. 55
1.5
Complex Numbers and Functions
. 57
Euler's Formula and Its Consequences
. 58
A New View on Trigonometric Functions
. 61
Euler's Product for the Sine Function
. 62
Exercises
. 66
1.6
Continued Fractions
. 68
Origins
. 68
Convergents
. 71
Irrationality
. 76
Exercises
. 78
Chapter II Differential and Integral Calculus
11.1
The Derivative
. 81
The Derivative
. 81
Differentiation Rules
. 84
Parametric Representation and Implicit Equations
. 88
Exercises
. 89
11.2 Higher Derivatives and Taylor Series
. 91
The Second Derivative
. 91
De Conversione Functionum in
Series
. 94
Exercises
. 97
11.3 Envelopes and Curvature
. 98
Envelope of a Family of Straight Lines
. 98
The Caustic of a Circle
. 99
Envelope of Ballistic Curves
. 101
Curvature
. 101
Exercises
. 105
11.4 Integral Calculus
. 107
Primitives
. 107
viii Contents
Applications
.109
Integration
Techniques
.112
Taylor's
Formula
with Remainder
.116
Exercises
.117
II.5 Functions with Elementary Integral
.118
Integration of Rational Functions
.118
Useful Substitutions
.123
Exercises
.125
П.6
Approximate Computation of Integrals
.126
Series Expansions
.126
Numerical Methods
.128
Asymptotic Expansions
.131
Exercises
.132
11.7 Ordinary Differential Equations
.134
Some Types of
Integrable
Equations
.139
Second Order Differential Equations
.140
Exercises
.143
11.8 Linear Differential Equations
.144
Homogeneous Equation with Constant Coefficients
.145
Inhomogeneous Linear Equations
.148
Cauchy's Equation
.152
Exercises
.152
11.9 Numerical Solution of Differential Equations
.154
Euler's Method
.154
Taylor Series Method
.156
Second Order Equations
.158
Exercises
.159
11.10
The Euler-Maclaurin Summation Formula
.160
Euler's Derivation of the Formula
.160
De
Usu
Legitimo
Formulae Summatoriae Maclaurinianae
.163
Stirling's Formula
.165
The Harmonic Series and Euler's Constant
.167
Exercises
.169
Chapter III Foundations of Classical Analysis
111.1
Infinite Sequences and Real Numbers
.172
Convergence of a Sequence
.172
Construction of Real Numbers
.177
Monotone Sequences and Least Upper Bound
.182
Accumulation Points
.184
Exercises
.185
111.2 Infinite Series
.188
Criteria for Convergence
.189
Absolute Convergence
.192
Double Series
.195
The Cauchy Product of Two Series
.197
Exchange of Infinite Series and Limits
.199
Exercises
.200
111.3 Real Functions and Continuity
.202
Continuous Functions
.204
The Intermediate Value Theorem
.206
The Maximum Theorem
.206
Monotone and Inverse Functions
.208
Limit of a Function
.209
Exercises
.210
Contents ix
111.4
Uniform Convergence and Uniform Continuity
.213
The Limit of a Sequence of Functions
.213
Weierstrass's Criterion for Uniform Convergence
.216
Uniform Continuity
.217
Exercises
.220
111.5 The Riemann Integral
.221
Definitions and Criteria of Integrability
.221
Integrable
Functions
.226
Inequalities and the Mean Value Theorem
.228
Integration of Infinite Series
.230
Exercises
.232
111.6 Differentiable Functions
.235
The Fundamental Theorem of Differential Calculus
.239
The Rules of
de ĽHospital
.242
Derivatives of Infinite Series
.245
Exercises
.246
111.7
Power Series and Taylor Series
.248
Determination of the Radius of Convergence
.249
Continuity
.250
Differentiation and Integration
.251
Taylor Series
.252
Exercises
.255
111.8 Improper Integrals
.257
Bounded Functions on Infinite Intervals
.257
Unbounded Functions on a Finite Interval
.260
Euler's Gamma Function
.261
Exercises
.262
111.9 Two Theorems on Continuous Functions
.263
Continuous, but Nowhere Differentiable Functions
.263
Weierstrass's Approximation Theorem
.265
Exercises
.269
Chapter IV Calculus in Several Variables
IV.l Topology of n-Dimensional Space
.273
Distances and Norms
.273
Convergence of Vector Sequences
.275
Neighborhoods, Open and Closed Sets
.278
Compact Sets
.283
Exercises
.285
IV.2 Continuous Functions
.287
Continuous Functions and Compactness
.289
Uniform Continuity and Uniform Convergence
.290
Linear Mappings
.293
Hausdorff's Characterization of Continuous Functions
.294
Integrals with Parameters
.297
Exercises
.298
IV.3 Differentiable Functions of Several Variables
.300
Differentiability
.302
Counter-Examples
.304
A Geometrical Interpretation of the Gradient
.305
The Mean Value Theorem
.308
The Implicit Function Theorem
.309
Differentiation of Integrals with Respect to Parameters
.311
Exercises
.313
IV.4 Higher Derivatives and Taylor Series
.316
Taylor Series for Two Variables
.319
χ
Contents
Taylor
Series
for n Variables
.320
Maximum and Minimum Problems
.323
Conditional Minimum
(Lagrange
Multiplier)
.325
Exercises
.328
IV.5 Multiple Integrals
.330
Double Integrals over a Rectangle
.330
Null Sets and Discontinuous Functions
.334
Arbitrary Bounded Domains
.336
The Transformation Formula for Double Integrals
.338
Integrals with Unbounded Domain
.345
Exercises
.347
Appendix: Original Quotations
.351
References
.358
Symbol Index
.369
Index
.371 |
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| author | Hairer, Ernst 1949- Wanner, Gerhard |
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| ctrlnum | (OCoLC)191759999 (DE-599)DNB986297763 |
| dewey-full | 515 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515 |
| dewey-search | 515 |
| dewey-sort | 3515 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| era | Geschichte gnd |
| era_facet | Geschichte |
| format | Book |
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| title_full_unstemmed | Analysis by its history E. Hairer ; G. Wanner |
| title_short | Analysis by its history |
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